Volterra equations (1st and 2nd kind) are a common solution to many electrochemical problems. I have solutions for these in my book "Simulating Electrochemical Reactions with Mathematica" but the short answer is to adopt a finite difference scheme using what in the echem literature is known as Huber's method. The solution can be obtained very rapidly using standard approaches in Mma.
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Mike HoneychurchApr 24 '12 at 4:42

@MikeHoneychurch I looked at your book an Amazon, very interesting! Maybe you can give a brief outline as an answer (wouldn't want to plagiarize your book, of course). Although this question lacks specificity, the general techniques (or references) might be of interest...
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JensApr 24 '12 at 5:36

@Jens I went and had a look at the relevant parts of the book prior to posting. Because of the way the thing is laid out it didn't seem like something that could readily be chopped up and displayed here to readily answer the Ops question. If the OP is familiar with discretizing an integral equation then he should be able to work through the problem. Next step is to code it. I'll see if I can work through some generic examples.
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Mike HoneychurchApr 24 '12 at 7:33

Yes, this works well for non-singular kernels. That was initially why I thought this question could be considered a duplicate of How to invert an integral equation. But which method works best here can't really be decided from the facts given in the question...
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JensApr 24 '12 at 17:19

There are many sources that give practical advice on how to program such an integral equation. So I went a different route and asked myself how to get a solution by using the most literal application of the defining equation.

By that I mean no explicit discretization, keeping the integral. That can be done using the assumption that the fixed-point theorem holds for the given equation. That theorem means that if I replace $P(t')$ under the integral by the entire expression for $P(t)$, with $t\to t'$, and keep doing that with all the $P(...)$ that appear, the result will converge.

Here is a variant from one response that I cobbled together to illustrate your case. I chose the kernel function (your R_0) to be
e^(-x) + x sin(x) - x cos(x^2).
I attempt to solve the integral equation over the range from 0 to 1, iterating to get improvements. We start by setting the initial value of P(x) equal to kernel(x). We'll do an iteration by evaluation the integral with this initial P(x), at 100 points in (0,1). We use those to make an interpolating function, and that becomes the new approximation to P(x). I did four iterations of this process.

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